Recent advances in permafrost modelling - Department of Geography

PERMAFROST AND PERIGLACIAL PROCESSES 

Permafrost and Periglac. Process. 19: 137–156 (2008) 

Published online in Wiley InterScience 

(www.interscience.wiley.com) DOI: 10.1002/ppp.615 

Recent Advances in Permafrost Modelling 

Daniel Riseborough , 1 * Nikolay Shiklomanov, 2 Bernd Etzelmüller, 3 Stephan Gruber 4 and Sergei Marchenko 5 

1 Geological Survey of Canada, Ottawa, Canada 

2 Department of Geography, University of Delaware, Newark, DE, USA 

3 Department of Geosciences, Physical Geography, University of Oslo, Oslo, Norway 

4 Department of Geography, University of Zürich, Zürich, Switzerland 

5 Geophysical Institute, University of Alaska, Fairbanks, AK, USA 

ABSTRACT 

This paper provides a review of permafrost modelling advances, primarily since the 2003 permafrost 

conference in Zürich, Switzerland, with an emphasis on spatial permafrost models, in both arctic and high 

mountain environments. Models are categorised according to temporal, thermal and spatial criteria, and 

their approach to defining the relationship between climate, site surface conditions and permafrost status. 

The most significant recent advances include the expanding application of permafrost thermal models 

within spatial models, application of transient numerical thermal models within spatial models and 

incorporation of permafrost directly within global circulation model (GCM) land surface schemes. Future 

challenges for permafrost modelling will include establishing the appropriate level of integration required 

for accurate simulation of permafrost-climate interaction within GCMs, the integration of environmental 

change such as treeline migration into permafrost response to climate change projections, and parameterising 

the effects of sub-grid scale variability in surface processes and properties on small-scale (large 

area) spatial models. Copyright # 2008 John Wiley & Sons, Ltd. 

KEY WORDS: permafrost; models; geothermal; mountains; spatial models 

INTRODUCTION 

Permafrost (defined as ground where temperatures have 

remained at or below 08C for a period of least two 

consecutive years) is a key component of the cryosphere 

through its influence on energy exchanges, hydrological 

processes, natural hazards and carbon budgets — and 

hence the global climate system. The climatepermafrost 

relation has acquired added importance 

with the increasing awareness and concern that rising 

temperatures, widely expected throughout the next 

century, may particularly affect permafrost environments. 

The Intergovernmental Panel on Climate Change 

* Correspondence to: Daniel Riseborough, Geological Survey 

of Canada, 601 Booth Street, Ottawa, Ontario, Canada K1A 

0E4. E-mail: drisebor@nrcan.gc.ca 

Copyright # 2008 John Wiley & Sons, Ltd. 

(1990) has advocated that research should be directed 

towards addressing the climate-permafrost relation, 

including the effects of temperature forcing from 

climatic variation, local environmental factors such as 

snow and vegetation, and surficial sediments or bedrock 

types. Permafrost has been identified as one of six 

cryospheric indicators of global climate change within 

the international framework of the World Meteorological 

Organization (WMO) Global Climate Observing 

System (Brown et al., 2008). 

This review gives an overview of permafrost 

modelling advances, primarily since the 2003 permafrost 

conference in Zurich, Switzerland, with an 

emphasis on spatial permafrost models, in both arctic 

and high mountain environments. Many models have 

been developed to predict the spatial variation of 

permafrost thermal response to changing climate 

Received 3 January 2008 

Revised 19 March 2008 

Accepted 19 March 2008

138 D. Riseborough et al. 

conditions at different scales, including conceptual, 

empirical and process-based models. The permafrost 

models considered in this paper are those that define the 

thermal condition of the ground, based on some 

combination of climatic conditions and the properties 

of earth surface and subsurface properties. Particular 

emphasis is given to models used in spatial applications. 

MODELLING BACKGROUND 

A model is a conceptual or mathematical representation 

of a phenomenon, usually conceptualised as a 

system. It provides an idealised framework for logical 

reasoning, mathematical or computational evaluation 

as well as hypothesis testing. Explicit assumptions 

about or simplifications of the system may be part of 

model development; how these decisions affect the 

utility of the model will depend on whether the model 

serves its intended purpose with satisfactory accuracy. 

Rykiel (1996) suggests that model ‘validation’ can be 

separated into evaluations of theory, implementation 

and data; while some statistical tests can be performed 

to measure differences between model behaviour and 

the behaviour of the system, ‘validation’ ultimately 

depends on whether the model and its behaviour are 

reasonable in the judgement of knowledgeable people. 

The value of a model depends on its usefulness for a 

given purpose and not its sophistication. Simple models 

can be more useful than models which incorporate many 

processes, especially when data are limited. Choosing 

the appropriate point along the continuum of model 

complexity depends on data availability, modelling 

(usually computer) resources and the questions under 

investigation. The data needed to obtain credible results 

usually increase with increasing model complexity and 

with the number of processes that are represented. In 

mountain environments, lateral variations of surface and 

subsurface conditions as well as micro-climatology 

are far greater than in lowland environments. 

Process-based permafrost models determine the 

thermal state of the ground based on principles of heat 

transfer, and can be categorised using temporal, thermal 

and spatial criteria. Temporally, models may define 

equilibrium permafrost conditions for a given annual 

regime (equilibrium models), or they may capture 

the transient evolution of permafrost conditions from 

some initial state to a modelled current or future state 

(transient models). 

Thermally, simple models may define the presence 

or absence of permafrost, active-layer depth, or mean 

annual ground temperature, based on empirical and 

statistical relations or application of so-called equilibrium 

models utilising transfer functions between 

atmosphere and ground. Numerical models (finiteelement 

or finite-difference models) may define the 

annual and longer term progression of a deep-ground 

temperature profile (transient models). The influence 

of surface energy exchange on subsurface conditions 

may be defined by simplified equations using a few 

parameters (such as freezing/thawing indices and n 

factors, or air temperature and snow cover in 

equilibrium models), or as a fully explicit energy 

balance, requiring data for atmospheric conditions. 

Spatially, models may define conditions at a single 

location (either as an index or a one-dimensional 

vertical temperature profile), along a two-dimensional 

transect, or over a geographic region. Most geographic/ 

spatial permafrost models today are collections of 

modelled point locations, with no lateral heat flow 

between adjacent points. Points in such spatial models 

behave as one-dimensional models, so that small-scale 

spatial variability in soil properties and snow cover is 

not considered. This is a central problem for mountain 

areas where the effects of slope, aspect and elevation 

must be considered at regional or local scales. Thus, in 

mountainous regions the scale of variation in these 

factors requires alternative approaches to spatial 

modelling. 

Heat Flow Theory 

Basic heat flow theory is described in some detail in 

textbooks, such as Carslaw and Jaeger (1959), 

Williams and Smith (1989) and Lunardini (1981). 

The equation for heat flow under transient conditions 

forms the basis of all geothermal models. 

C @T 

@t ¼ k @2T @z2 (1) 

Definitions of equation symbols are given in Table 1. 

Two exact analytical models derived for a semiinfinite 

and isotropic half-space using equation 1 are: the 

harmonic solution, describing ground temperatures at 

any time t and depth z below a ground surface 

experiencing sinusoidal temperature variations: 

Tz;t ¼ T þ As e z 

p 

sin 2pt 

P 

ffiffiffiffiffiffiffiffi 

p=aP 

pffiffiffiffiffiffiffiffiffiffiffi 

z p=aP 

(2) 

and the step change solution, describing changes to 

ground temperatures at any time t and depth z below a 

ground surface following a step change in ground 

Copyright # 2008 John Wiley & Sons, Ltd. Permafrost and Periglac. Process., 19: 137–156 (2008) 

DOI: 10.1002/ppp

Table 1 Symbols used in equations. 

A s 

¼ annual temperature amplitude at soil 

surface, o C 

c ¼ specific heat, Jkg 1 

C ¼ volumetric heat capacity, Jm 3 

IFA ¼ seasonal air freezing index, 8Cs 

IFS ¼ seasonal ground surface freezing index, 8Cs 

ITA ¼ seasonal air thawing index, 8Cs 

ITS ¼ seasonal ground surface thawing index, 8Cs 

L ¼ volumetric latent heat of fusion, Jm 3 

nF ¼ surface freezing n-factor 

¼ surface thawing n-factor 

nT 

P ¼ period of the temperature wave, s 

P sn 

¼ period of the temperature wave, adjusted for 

snow melt, s 

t ¼ time, s 

T ¼ mean annual temperature, 8C 

T0 ¼ initial soil temperature, 8C 

TF ¼ fusion temperature, 8C 

TS ¼ surface temperature, 8C 

TTOP ¼ temperature at top of perennially 

frozen/unfrozen ground, 8C 

Tz ¼ mean annual temperature at the depth of 

seasonal thaw (equivalent to TTOP), 8C 

T z,t 

¼ temperature at depth z at time t, 8C 

x ¼ volume fraction 

X ¼ depth of thaw, m 

z ¼ depth, m 

uU 

r 

¼ volumetric unfrozen water content 

¼ density, kg m 3 

a ¼ l = C ¼ thermal diffusivity m 2 s 1 

l ¼ thermal conductivity, Wm 1 K 1 

Subscripts for a, l, r, c, C and n: 

a ¼ apparent 

i ¼ subscript identifying component 

(mineral, ice, water, etc.) 

T ¼ thawed or thawing 

F ¼ frozen or freezing 

surface temperature: 

DTz;t ¼ DTS erfc 

z 

2 ffiffiffiffiffi p (3) 

a t 

(erfc is the complementary error function) 

Permafrost models are a subset of a more general 

class of geothermal models. In permafrost models, 

ground freezing and thawing are central in determining 

the important variables and parameters of which 

the model is comprised. Equations 2 and 3 form the 

basis of analyses of ground temperatures outside of 

permafrost regions, and are most useful where 

Recent Advances in Permafrost Modelling 139 

freezing and thawing of significant amounts of soil 

moisture do not occur. Exact analytical models of the 

thermal behaviour of the ground when freezing or 

thawing occurs are limited to a few idealised 

conditions (Lunardini, 1981). Real-world conditions 

such as seasonal variation in the ground surface 

temperature, accumulation and ablation of snow 

cover, and temperature dependent thermal properties, 

are beyond the capacity of these exact models. Two 

paths move beyond this impasse: approximate 

analytical models developed by making simplifying 

assumptions, or numerical techniques employed to 

solve more complex problems with the acceptance of 

limited error. 

For ground that undergoes freezing and thawing, the 

release and absorption of the latent heat of fusion of 

the soil water dominate heat flow, although the 

temperature-dependence of thermal conductivity is 

also important. Accounting for latent heat is usually 

achieved by subsuming its effect in the heat capacity 

term in equation 1. 

The Stefan Model 

Ca ¼ X xir ici þ L @uu= @T 

(4) 

The analytical equation most widely employed in the 

formulation of permafrost models is the Stefan 

solution to the moving freezing (or thaw) front. When 

diffusive effects are small relative to the rate of frost 

front motion and the initial temperature of the ground 

is close to 08C, the exact equation for the moving 

phase change boundary can be simplified to a form of 

the Stefan solution using accumulated ground surface 

degree-day total I (either the freezing index I F or 

thawing index IT) (Lunardini, 1981): 

rffiffiffiffiffiffiffi 

2lI 

X ¼ 

L 

(5) 

The form of Stefan solution represented by equation 

(5) is widely used for spatial active-layer characterisation 

by estimating soil properties (‘edaphic 

parameters’) empirically, using summer air temperature 

records and active-layer data obtained from 

representative locations (e.g. Nelson et al., 1997; 

Shiklomanov and Nelson, 2003; Zhang T. et al., 

2005). 

Carlson (1952, based on Sumgin et al., 1940) used 

equation 5 as the origin of a simple model for presence 

of permafrost, based on the notion that permafrost will 


DOI: 10.1002/ppp


be present where predicted winter season freezing 

exceeds predicted summer season thaw, so that 

permafrost exists where kFIFS > kTITS. Nelson and 

Outcalt (1987) derived the Frost Index model on this 

relationship. While the Frost Index model has been 

used extensively (e.g. Anisimov and Nelson, 1996), 

the Kudryavstev model has become more common in 

recent studies. 

The Kudryavtsev Model 

An alternative solution to the Stefan problem was 

proposed by Kudryavtsev et al. (1974) (Figure 1A) for 

estimating maximum annual depth of thaw propagation 

and the mean annual temperature at the base of 

the active layer Tz (equivalent to the temperature at the 

top of permafrost, or TTOP, described below): 

where 

h i 

Az ¼ As Tz 

ln AsþL=2CT 

TzþL=2CT 

and 

Zc ¼ 2ðAs TzÞ 

L 

2CT 

qffiffiffiffiffiffiffiffiffiffiffiffi 

l Psn CT 

p 

2Az CT þ L 

The mean annual temperature at the depth of seasonal 

thaw (permafrost surface) can be calculated as: 

Tz ¼ 

Zthaw ¼ 

2ðAs TzÞ 

0:5Ts ðlF þ lTÞþAs lF lT 

p 

l 

l ¼ lF; if numerator < 0 

lT; if numerator > 0: 

h qffiffiffiffiffiffiffiffiffiffiffiffi 

i 

Ts 

As 

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

þ 

lT Psn CT 

p 

arcsin Ts 

As þ 

2Az CT þ L 

1 p 2 

A 2 s 

(7) 

Kudryavtsev’s equations were derived assuming a 

periodic steady state with phase change (L > 0) 

(Kudryavtsev et al., 1974). Romanovsky and Osterkamp 

(1997) indicate that equations 6 and 7 can be 

applied on an annual basis. Extensive validation of 

Kudryavtsev’s equations using empirical data from the 

North Slope of Alaska indicate that they provide more 

accurate estimates of annual maximum seasonal thaw 

depth than the Stefan equation (Romanovsky and 

Osterkamp, 1997; Shiklomanov and Nelson, 1999). 

Kudryavtsev’s model is used extensively at regional 

(Shiklomanov and Nelson, 1999; Sazonova and 

Romanovsky, 2003; Stendel et al., 2007) and 

circum-arctic (Anisimov et al., 1997) scales. 

N Factors 

Freezing and thawing n factors relate ground surface 

temperature to air temperature as an empirical 

alternative to the energy balance (Lunardini, 1978). 

N factors are applied to seasonal degree-day totals 

(FDD or IF for freezing; TDD or IT for thawing), 

calculated as the accumulated departure of mean daily 

ð2Az CT ZcþL ZcÞ L 

2Az CT ZcþL Zþð2Az CT þLÞ 

temperature above (or below) 08C, so that equation 5 

can be applied using air temperature data. 

nT ¼ ITS 

; nF ¼ 

ITA 

IFS 

IFA 

(8) 

N factors account in a lumped form for the complex 

processes within the atmosphere-soil system and, as 

such, n factors will vary for a given location. Shur and 

Slavin-Borovskiy (1993) found that site-specific n 

factors are stable, with interannual changes less than 

10 per cent in continental arctic areas. In mountainous 

regions, however, n factor variations can be much 

higher, especially during winter due to interannual 

variation of snow cover (e.g. Juliussen and Humlum, 

2007). This variability might be even more accentuated 

in maritime mountainous areas (e.g. Etzelmüller 

et al., 2007, 2008). In low-topography, 

continental areas n factors can be correlated with 

surface cover and extrapolated over larger areas (e.g. 

Duchesne et al., 2008). 

The TTOP Model 

pffiffiffiffiffiffiffi 

l Psn 

p C pT 

ffiffiffiffiffiffiffi 

l Psn 

p C T 

(6) 

The TTOP model (Smith and Riseborough, 1996) 

(Figure 1B) estimates the mean annual temperature at 

the top of perennial frozen/unfrozen soil by combining a 

model (Romanovsky and Osterkamp, 1995) of the 

thermal offset effect (in which the mean annual 


DOI: 10.1002/ppp

Figure 1 (A) Conceptual drawing of the Kudryavtsev model; (B) 

temperature at the top of permafrost (TTOP) model conceptual 

schematic profile of mean annual temperature through the lower 

atmosphere, active layer and upper permafrost. 

temperature shifts to lower values in the active layer 

because of the difference between frozen and 

thawed thermal conductivities of the soil) with freezing 

and thawing indices in the lower atmosphere linked to 

values at the ground surface using n factors (equation 8): 

TTOP ¼ nTlT ITA nFlF IFA 

; 

kFP 

TTOP < 0 

TTOP ¼ nTlT ITA nFlF IFA 

; 

lTP 

TTOP > 0 

(9) 

Applications of the TTOP model have depended on 

empirically derived n factors (Wright et al., 2003; 

Juliussen and Humlum, 2007), although Riseborough 


(2004) attempted to develop a physically based 

n-factor parameterisation of the effect of snow cover. 

Statistical-Empirical Models and 

Multiple-Criteria Analysis 

Statistical-empirical permafrost models relate permafrost 

occurrences to topoclimatic factors (such as 

altitude, slope and aspect, mean air temperature, or 

solar radiation), which can easily be measured or 

computed. This type of model is widely used in 

mountain permafrost studies (see also Hoelzle et al., 

2001), assumes equilibrium conditions and usually 

relies on basal temperature of the snow cover (BTS), 

temperatures measured with miniature data loggers, 

geophysical investigations, or the existence of specific 

landforms (rock glaciers) as evidence of permafrost 

occurrence. In the early 1990s, the availability of 

Geographical Information Systems (GIS) allowed the 

first estimation and visualisation of the spatial 

distribution of permafrost in steep mountains. PER- 

MAKART (Keller, 1992) uses an empirical topographic 

key for permafrost distribution and in the 

PERMAMAP approach of Hoelzle and Haeberli 

(1995), BTS measurements, which are associated 

with the presence or absence of permafrost, are 

statistically related to mean annual air temperature 

(MAAT, estimated using nearby climate station 

records) and potential direct solar radiation during 

the snow-free season, calculated using digital 

elevation models (DEMs) (Funk and Hoelzle, 

1992). The challenges with these approaches include 

the often indirect nature of the driving information on 

permafrost, the strong interannual variability of BTS, 

and the need to recalibrate for different environments. 

Statistical-empirical models usually estimate mean 

ground temperatures or provide measures of permafrost 

probability. 

For small-scale mapping of large mountain areas, 

MAAT is often used as the sole predictor of permafrost 

occurrence, using thresholds based on field measurements. 

This approach involves the spatial interpolation 

of air temperature patterns from a common reference 

elevation (usually sea level) and the subsequent 

calculation of the near-surface MAAT using a DEM 

and a (sometimes spatially variable) lapse rate. Studies in 

southern Norway and Iceland have shown that a MAAT 

of 3to 48C is a good estimate for the regional limit of 

the lower mountain permafrost boundary (Etzelmüller 

et al., 2003, 2007). These threshold values are normally 

estimated based on ground surface temperature data 

from several locations, or other proxy information such 

as landforms. The resulting mountain permafrost 

distributions are then compared with more local-scale 


DOI: 10.1002/ppp


observations or maps, confirming the overall spatial 

pattern for Scandinavia (e.g. Isaksen et al., 2002; 

Heggem et al., 2005; Etzelmüller et al., 2007; Farbrot 

et al., 2008). 

For more remote areas or regions with sparse data 

on permafrost indicators, multi-criteria approaches 

within a GIS framework have been applied to generate 

maps of ‘permafrost favorability’. Here, scores were 

derived for single factors (elevation, topographic 

wetness, potential solar radiation, vegetation) based 

on simple logistic regression or basic process understanding, 

with the sum of the derived probabilities 

used as a measure of permafrost favourability in a 

given location (Etzelmüller et al., 2006). 

Numerical Models 

The limitations of equilibrium models have spurred 

the recent adaptation of transient numerical simulation 

models in spatial applications. Numerical models are 

flexible enough to accommodate highly variable 

materials, geometries and boundary conditions. Most 

thermal models for geoscience applications are 

implemented by simulating vertical ground temperature 

profiles in one dimension employing a finitedifference 

or finite-element form of equation 1, 

usually following a standard procedure: 

1. Define the modelled space: set a starting point in 

time, and upper and lower boundaries. 

2. Divide continuous space into finite pieces (a grid of 

nodes or elements) and continuous time into finite 

time steps. 

3. Specify the thermal properties of the soil materials. 

4. Specify the temperature or heat flow conditions as a 

function of time (i.e. for each time step) for the 

upper and lower boundaries. 

5. Specify an initial temperature for every point in the 

profile. 

6. For each time step after the starting time, the new 

temperature profile is calculated, based on the 

combination of thermal properties, antecedent 

and boundary conditions. 

While the use of numerical models allows the 

accommodation of heterogeneity in both space and 

time, it also gives rise to the problem of actually 

supplying spatial data fields of material properties 

and initial conditions. 

Upper Boundary Conditions 

Upper boundary temperature conditions in numerical 

models can be specified in various ways. Temperatures 

may be specified for the ground surface directly or 

using n factors (e.g. Duchesne et al., 2008); 

alternately, when snow cover is present the temperature 

at the snow surface may be specified so that 

ground surface temperature is determined by heat flow 

between the snow and the ground (e.g. Oelke and 

Zhang, 2004). The most elaborate method of 

establishing the surface boundary temperature is by 

calculation of the surface energy balance to determine 

the equilibrium temperature (Budyko, 1958) at the 

snow or ground surface (e.g. Zhang et al., 2003). 

Surface energy-balance models generally employ a 

radiation balance with partitioning of atmospheric 

sensible and latent heat using aerodynamic theory. The 

earlier generation of permafrost models generally 

employed data that were collected locally for 

site-specific application, including incoming solar 

radiation, wind speed and air temperature (e.g. Outcalt 

et al., 1975; Ng and Miller, 1977; Mittaz et al., 2000). 

Spatial permafrost modelling at regional (Hinzman 

et al., 1998; Chen et al., 2003) and continental 

(Anisimov, 1989; Zhang et al., 2007) scales has 

required a less site-specific approach to the energy 

balance, with short-wave radiation attenuated through 

the atmosphere, wind fields modified by local 

conditions based on satellite-derived leaf area indices, 

etc., often with climatic conditions obtained from 

GCM output (e.g. Sushama et al., 2007). In mountain 

areas, extreme spatial variability requires the parameterisation 

of the influence of topography on surface 

micro-climatology for the derivation of spatially 

distributed information about temperature and snow 

cover (cf. Stocker-Mittaz et al., 2002; Gruber, 2005). 

SPATIAL MODELS 

Application of the types of models described in the 

previous sections to the simulation and prediction of 

permafrost distribution at continental, regional and 

local scales is discussed in the following sections. 

Mountain permafrost models are discussed in a separate 

section, as is recent work incorporating permafrost into 

General Circulation Model (GCM) land surface 

schemes. 

Continental/circumpolar 

Due to the strong dependence of permafrost conditions 

on regional climate, geocryological spatial modelling 

has been dominated by national- to circumpolar-scale 

(i.e. small scale) studies in which broad spatial 

patterns can be related to a few readily available 


DOI: 10.1002/ppp

climatic parameters. The empirical methods frequently 

used to assess permafrost distribution at these 

scales implicitly assume that the thermal regime of 

near-surface permafrost is determined by modern 

climatic conditions. The use of macro-scale patterns of 

climatic parameters to infer the configuration of 

permafrost zones was first proposed by G. Wild 

(Shiklomanov, 2005), who was able to correlate the 

southern boundary of permafrost with the 28C 

isotherm of MAAT (Wild, 1882). Such simple 

empirical relations were frequently used throughout 

the twentieth century for permafrost regionalisation 

(Heginbottom, 2002), although they do not account for 

the thermal inertia of permafrost. 

Equilibrium Models. 

The Frost Number model was applied successfully 

to central Canada (Nelson, 1986) and continental 

Europe (Nelson and Anisimov, 1993) for modern 

climatic conditions. Calculated boundaries of continuous 

and discontinuous permafrost were in satisfactory 

agreement with existing geocryological maps. 

The Frost Number was used with an empirical 

scenario derived from palaeoanalogues and with 

output from several GCMs to predict the future 

distribution of ‘climatic’ permafrost in the northern 

hemisphere (Anisimov and Nelson, 1996; Anisimov 

and Nelson, 1997). An alternative approach was used 

by Smith and Riseborough (2002) to evaluate the 

conditions controlling the limits and continuity of 

permafrost in the Canadian Arctic by means of the 

TTOP model. Using a model derived from the TTOP 

model, based primarily on the roles of snow and soil 

thermal properties on the thermal effect of the winter 

snow cover, Riseborough (2004) produced maps of the 

southern boundary of permafrost in Canada for a range 

of substrate conditions. 

Several variations of the Kudryavtsev model have 

been used with GIS technology to calculate both 

active-layer thickness and mean annual ground temperatures 

at circum-arctic scales (Anisimov et al., 

1997). The thermal and physical properties of snow, 

vegetation, and organic and mineral soils were fixed 

both spatially and temporally. These properties were 

varied stochastically within a range of published data for 

each grid cell, producing a range of geographically 

varying active-layer estimates. The final active-layer 

field was produced by averaging of intermediate results. 

The model was extensively used to evaluate the 

potential changes in active-layer thickness under 

different climate change scenarios at circumpolar 

(Anisimov et al., 1997) and continental (Anisimov 

and Reneva, 2006) scales, to assess the hazard potential 

associated with progressive deepening of the active 


layer (Nelson et al., 2001, 2002; Anisimov and Reneva, 

2006) and to evaluate the emission of greenhouse gases 

from the Arctic wetlands under global warming 

conditions for Russian territory (Anisimov et al., 

2005; Anisimov and Reneva, 2006). 

Numerical Models. 

A one-dimensional finite-difference model for heat 

conduction with phase change and a snow routine 

(Goodrich, 1978, 1982) has been adapted at the National 

Snow and Ice Data Center (NSIDC) to simulate soil 

freeze/thaw processes at regional or hemispheric scales 

(Oelke et al., 2003, 2004; Zhang T. et al., 2005). The 

NSIDC permafrost model requires gridded fields of 

daily meteorological parameters (air temperature, 

precipitation), snow depth and soil moisture content, 

as well as spatial representation of soil properties, land 

cover categories, and DEMs to evaluate the daily 

progression of freeze/thaw cycles and soil temperature 

at specified depth(s) over the modelling domain. Initial 

soil temperatures are prescribed from available empirical 

observations associated with permafrost classification 

from the International Permafrost Association’s 

Circum-Arctic Permafrost Map (Brown, 1997; Zhang 

et al., 1999), with a grid resolution of 25 km 25 km. 

The numerical model has been shown to provide 

excellent results for active-layer depth and permafrost 

temperatures (Zhang et al., 1996; Zhang and Stamnes, 

1998) when driven with well-known boundary conditions 

and forcing parameters at specific locations. 

Several numerical models have been developed 

recently for simulating permafrost evolution at 

continental scales. The Main Geophysical Observatory 

(St Petersburg) model was developed for Russian 

territory, using principles similar to those applied at 

the NSIDC but differing in computational details and 

parameterisations. In particular, the model was 

designed to be driven by GCM output to provide a 

substitute for unavailable empirical observations 

(Malevsky-Malevich et al., 2001; Molkentin et al., 

2001). 

Zhang et al. (2006) developed the Northern 

Ecosystem Soil Temperature (NEST) numeric model 

to simulate the evolution of the ground thermal regime 

of the Canadian landmass since the Little Ice 

Age (1850) (Figure 2). The model explicitly considered 

the effects of differing ground conditions, including 

vegetation, snow, forest floor or moss layers, peat layers, 

mineral soils and bedrock. Soil temperature dynamics 

were simulated with the upper boundary condition (the 

ground surface or snow surface when snow is present) 

determined by the surface energy balance and the lower 

boundary condition (at a depth of 120 m) defined by the 

geothermal heat flux (Zhang et al., 2003). The NEST 


DOI: 10.1002/ppp


Figure 2 Modelled changes in depth to the permafrost base from 

the 1850s to the 1990s, simulated using the Northern Ecosystem Soil 

Temperature numeric model (Zhang et al., 2006). 

model operates at half-degree latitude/longitude resolution 

and requires inputs relating to vegetation 

(vegetation type and leaf area index), ground conditions 

(thickness of forest floor and peat, mineral soil texture 

and organic carbon content, ground ice content, thermal 

conductivity of bedrock and geothermal heat flux) and 

atmospheric climate (air temperature, precipitation, 

solar radiation, vapour pressure and wind speed). 

Marchenko et al. (2008) developed University of 

Alaska Fairbanks-Geophysical Institute Permafrost 

Lab model Version 2 (UAF-GIPL 2.0), an Alaskaspecific, 

implicit finite-difference, numerical model 

(Tipenko et al., 2004). The formulation of the 

one-dimensional Stefan problem (Alexiades and 

Solomon, 1993; Verdi, 1994) makes it possible to 

use coarse vertical resolution without loss of 

latent-heat effects in the phase transition zone, even 

under conditions of rapid or abrupt changes in the 

temperature fields. Soil freezing and thawing follow 

the unfrozen water content curve in the model, 

specified for each grid point and soil layer, down to the 

depth of constant geothermal heat flux (typically 500 

to 1000 m). The model uses gridded fields of monthly 

air temperature, snow depth, soil moisture, and 

thermal properties of snow, vegetation and soil at 

0.58 0.58 resolution. Extensive field observations 

from representative locations characteristic of the 

major physiographic units of Alaska were used to 

develop gridded fields of soil thermal properties and 

moisture conditions. 

Non-linearity of sub-grid scale processes may lead to 

biased estimation of permafrost parameters when input 

data and computational results are averaged at the grid 

scale. While coarse spatial resolution is sufficient for 

many small-scale geocryological applications such as 

permafrost regionalisation, evaluation of spatial variations 

in near surface permafrost temperature and the 

active layer should account for more localised spatial 

variability. This problem is addressed to some extent by 

regional, spatial permafrost models. 

Local/regional 

In general the approaches to permafrost modelling 

at regional scale are similar to those at small 

geographical scale. However, regional, spatial permafrost 

models are applied in areas for which significant 

amounts of data are available, with the underlying 

principle to ‘stay close to the data’. Available 

observations allow comprehensive analysis of landscape-specific 

(vegetation, topography, soil properties 

and composition) and climatic (air temperature and 

its amplitude, snow cover, precipitation) characteristics 

that influence the ground thermal regime and 

provide realistic parameterisations for high-resolution 

modelling. 

Empirical Models. 

Using empirically derived landscape-specific edaphic 

parameters representing the response of the active layer 

and permafrost to both climatic forcing and local factors 

(soil properties, moisture conditions and vegetation), 

empirical spatial modelling was successfully applied to 

high-resolution spatial modelling of the annual thaw 

depth propagation over the 29 000 km 2 Kuparuk region 

in north-central Alaska (Nelson et al., 1997; Shiklomanov 

and Nelson, 2002) and the northern portion of 

west Siberia (Shiklomanov et al., 2008). Zhang T. et al. 

(2005) used a landscape-sensitivity approach in conjunction 

with Russian historic ground temperature 

measurements to evaluate spatial and temporal variability 

in active-layer thickness over several Russian 

arctic drainage basins. 

Equilibrium Models (Kudryavtsev, Stefan 

and TTOP). 

The Kudryavtsev approach was applied successfully 

over the Kuparuk region of north-central Alaska 

by Shiklomanov and Nelson (1999) for highresolution 

(1 km 2 ) characterisation of active-layer 

thickness and was used as the basis for the GIPL 1.0, 

an interactive GIS model designed to estimate the 

long-term response of permafrost to changes in 

climate. The GIPL 1.0 model has been applied to 

detailed analysis of permafrost conditions over two 

regional transects in Alaska and eastern Siberia 

(Sazonova and Romanovsky, 2003). In particular it 


DOI: 10.1002/ppp

was used to simulate the dynamics of active-layer 

thickness and ground temperature, both retrospectively 

and prognostically, using climate forcing from 

six GCMs (Sazonova et al., 2004). To refine its spatial 

resolution, the GIPL 1.0 model was used in conjunction 

with a regional climate model (RCM) to provide 

more realistic trends of permafrost dynamics over the 

east-Siberian transect (Stendel et al., 2007). 

Stefan-based methods range from straightforward 

computational algorithms requiring deterministic 

specification of input parameters (gridded fields of 

thawing indices, thermal properties of soil, moisture/ 

ice content and land cover characteristics, and terrain 

models) (Klene et al., 2001) to establishing empirical 

and semi-empirical relationships between variables 

based on comprehensive field sampling (Nelson et al., 

1997; Shiklomanov and Nelson, 2002). These 

methods were used for high-resolution mapping of 

the active layer in the Kuparuk region (Klene et al., 

2001) and for detailed spatial characterisation of 

active-layer thickness in an urbanised area in the 

Arctic (Klene et al., 2003). 

Wright et al. (2003) used the TTOP formulation for 

high-resolution (1 km 2 ) characterisation of permafrost 

distribution and thickness in the broader Mackenzie 

River valley, north of 608N. The TTOP model 

calibrated for the Mackenzie region using observations 

of permafrost occurrence and thickness at 154 

geotechnical borehole sites along the Norman Wells 

Pipeline provided good general agreement with 

currently available information. The results indicate 

that given adequate empirically derived parameterisations, 

the simple TTOP model is well suited for 

regional-scale GIS-based mapping applications and 

investigations of the potential impacts of climate 

change on permafrost. 

Numerical Models. 

Although empirical and equilibrium approaches 

currently dominate spatial permafrost modelling at a 

regional scale, several numerical models were adopted 

for regional applications to provide spatial-temporal 

evolution of permafrost parameters. 

A regional, spatial, numerical thermal model was 

developed by Hinzman et al. (1998) and applied to 

simulate active layer and permafrost processes over 

the Kuparuk region, the North Slope of Alaska, at 

1km 2 resolution. The model utilises a surface energy 

balance and subsurface finite-element formulations to 

calculate the temperature profile and the depth of thaw. 

Meteorological data were provided by a local highdensity 

observational network and surface and 

subsurface characteristics were spatially interpolated 

based on measurements collected in typical landform 

and vegetation units. 

Duchesne et al. (2008) use a one-dimensional 

finite-element heat conduction model integrated with 

a GIS to investigate the transient impact of climate 

change on permafrost over three areas of intensive 

human activity in the Mackenzie valley (Figure 3). 

The model uses extensive field survey data and 

existing regional maps of surface and subsurface 

characteristics as input parameters to predict permafrost 

distribution and temperature characteristics at 

high resolution (1 km 2 ), and facilitates transient 

modelling of permafrost evolution over selected time 

frames. Statistical validation of modelling results 

indicates a reasonable level of confidence in model 

performance for applications specific to the Mackenzie 

River valley. 

Mountains 


The lateral variability of surface micro-climate and 

subsurface conditions is far greater in mountains than 

in lowland environments. The processes that govern 

the existence and evolution of mountain permafrost 

can be categorised into the scales and process domains 

of climate, topography and ground conditions 

(Figure 4). At the global scale, latitude and global 

circulation patterns determine the distribution of cold 

mountain climates. These climatic conditions are 

modified locally by topography, influencing microclimate 

and surface temperature due to differences in 

ambient air temperature caused by elevation, differences 

in solar radiation caused by terrain shape and 

orientation, and differences in snow cover due to 

transport by wind and avalanches. The influence of 

topographically altered climate conditions on ground 

temperatures is further modified locally by the 

physical and thermal properties of the ground. 

Substrate materials with high ice content can 

significantly retard warming and permafrost degradation 

at depth, while, especially in mountainous 

terrain, coarse blocky layers promote ground cooling 

relative to bedrock or fine-grained substrates (Hanson 

and Hoelzle, 2004; Juliussen and Humlum, 2008). 

Conceptually, these three scales and process domains 

are useful in understanding the diverse influences on 

mountain permafrost characteristics and the differences 

between modelling approaches, although 

divisions between scales are not sharply defined. 

The overall magnitude of the effect of topography on 

ground temperature conditions can be as high as 158C 

within a horizontal distance of 1 km — comparable to 

the effect of a latitudinal distance of 1000 km in polar 

lowland areas. 


DOI: 10.1002/ppp


Figure 3 Near-surface permafrost temperature in the Fort Simpson region of Northwest Territories, Canada, simulated using a 

finite-element model with surface boundary conditions (n factors, subsurface properties) derived from satellite and map-based vegetation 

characteristics (Duchesne et al., 2008). 

Empirical-Statistical and Analytical 

Distributed Models. 

Lewkowicz and Ednie (2004) designed a BTS- and 

radiation-based statistical model in the Yukon Territory 

in Canada, using an extensive set of observations in pits 

for calibration and validation. Lewkowicz and Bonnaventure 

(2008) examined strategies for transferring 

BTS-based statistical models to other regions, with the 

aim of modelling permafrost probabilities over the 

extensive mountain areas of western Canada. Brenning 

Figure 4 Conceptual hierarchy of scales and process domains that influence ground temperature and permafrost conditions in mountain 

areas. The white disk in the two leftmost images refers to a location that is then depicted in the image to the right — and has its conditions 

further overprinted by the respective conditions at that scale. 


DOI: 10.1002/ppp

et al. (2005) have outlined statistical techniques for 

improving the definition of such models, as well as for 

designing effective measurement campaigns. 

A very simple alternative approach is to represent 

the influence of solar radiation on ground temperatures 

using a model of the form: MAGT ¼ MAAT 

þ a þ b PR, where PR is the potential direct shortwave 

radiation, MAGT is the mean annual ground 

temperature, and a, b are model constants. This 

approach is usually limited by the available data for 

MAGT but can be a valuable first guess in remote 

locations (cf. Abramov et al., 2008). 

DEM-derived topographic parameters and information 

derived from satellite images offer the potential 

for more accurate permafrost distribution modelling in 

remote mountainous areas. Heggem et al. (2006) 

estimated the spatial distribution of mean annual ground 

surface temperature and active-layer thickness based on 

measured ground surface temperatures in different 

landscape classes defined by topographic parameters 

(elevation, potential solar radiation, wetness index) and 

satellite image-derived factors (forest and grass cover). 

A sine function was fitted to the surface temperature 

measurements (parameterised as the mean annual 

temperature and amplitude), providing input for the 

Stefan equation. Unlike the statistical approaches, this 

allowed the spatial mapping of simulated ground 

surface temperature and active-layer thickness fields 

for changing temperature or snow cover. 

Juliussen and Humlum (2007) proposed a TTOPtype 

model adapted for mountain areas. Elevation 

change is described through air temperatures, while 

slope dependencies are parameterised by potential 

solar radiation as a multiplicative summer-thaw n 

factor, including parameterisation for convective flow 

in blocky material. While there are benefits to this 

approach (such as a broad base of experience in 

lowland areas), the multiplicative treatment of solar 

radiation in the n factors will lead to problems when 

used over large ranges in elevation and needs further 

study. For Iceland, Etzelmüller et al. (2008) used the 

TTOP-modelling approach to estimate the influence of 

snow cover on permafrost existence at four mountain 

sites. The results were compared against transient heat 

flow modelling of borehole temperatures. Both studies 

show a high variability of winter n factors through the 

years of monitoring. 

Transient Models, Energy-balance Modelling and 

RCM-Coupling. 

In Scandinavia, one-dimensional transient models 

were applied to ground temperatures measured in 

permafrost boreholes in Iceland (Farbrot et al., 2007; 

Etzelmüller et al., 2008). 


The slope instability associated with the rapid 

thermal response of permafrost in steep bedrock 

slopes (cf. Gruber and Haeberli, 2007) has led to 

increased interest in measurements (Gruber et al., 

2003) and models (Gruber et al., 2004a, 2004b) of 

near-surface rock temperatures in steep terrain. The 

physics-based modelling (and validation) of temperatures 

in steep bedrock is an interesting subset of 

modelling the whole mountain cryosphere, because 

the influence of topography on micro-climate is 

maximised, while the influence of all other factors is 

minimal. The mostly thin snow cover on steep rock 

walls implies gravitational transport of snow and 

deposition at the foot of the slope. This is manifested 

in the occurrence of low-elevation permafrost as well 

as small glaciers that are entirely below the glacier 

equilibrium line altitude. A simple and fast algorithm 

for gravitational redistribution of snow (Gruber, 2007) 

is currently being tested in distributed energy-balance 

models (e.g. Strasser et al., 2007). 

Terrain geometry and highly variable upper boundary 

conditions determine the shape of the permafrost body, 

borehole temperature profiles (cf. Gruber et al., 2004c) 

and potential rates of permafrost degradation. Noetzli 

et al. (2007) have conducted detailed experiments with 

two- and three-dimensional thermal models, demonstrating 

that zones of very high lateral heat flux exist in 

ridges and peaks, and that these zones are subject to 

accelerated degradation as warming takes place from 

several sides (Noetzli et al., 2008). 

The one-dimensional model SNOWPACK (Bartelt 

and Lehning, 2002), originally developed to represent 

a highly differentiated seasonal snow cover, has been 

used in several pilot studies to investigate permafrost 

(Luetschg et al., 2004; Luetschg and Haeberli, 2005). 

In this model, mass and energy transport as well as 

phase change processes are treated with equal detail 

throughout all snow and soil layers, with water 

transported in a linear reservoir cascade. The 

distributed model Alpine3D (employing SNOW- 

PACK) can calculate or parameterise wind transport 

of snow, terrain-reflected radiation and snow structure 

(Lehning et al., 2006). 

Freeze-thaw processes combined with steep slopes 

produce significant fluctuations in water and ice content 

in the active layer of mountain permafrost. GEOTop 

(Zanotti et al., 2004; Bertoldi et al., 2006; Rigon et al., 

2006; Endrizzi, 2007) is a distributed hydrological 

model with coupled water and energy budgets that is 

specificallydesignedforuseinmountainareasandis 

currently being adapted to permafrost research. While 

the parameterisation, initialisation and validation of 

such complex models are demanding, the initial results 

of this research are promising. 


DOI: 10.1002/ppp


In contrast to high-latitude mountains and lowlands, 

the central Asian mid-latitudinal permafrost of the 

Tien Shan Mountains can be attributed exclusively to 

elevation. In the inner and eastern Tien Shan region the 

high level of solar radiation and high winds in 

combination with low atmospheric pressure and low 

humidity promote very intensive evaporation/sublimation 

in the upper ground and can lead to the 

formation of unexpectedly thick permafrost, especially 

within blocky debris (Haeberli et al., 1992; Harris, 

1996; Humlum, 1997; Harris and Pedersen, 1998; 

Delaloye et al., 2003; Gorbunov et al., 2004; Delaloye 

and Lambiel, 2005; Juliussen and Humlum, 2008; 

Gruber and Hoelzle, 2008). Spatial modelling of 

altitudinal permafrost in the Tien Shan and Altai 

mountains examined both permafrost evolution over 

time and permafrost dynamics at the local scale for 

specific sites within selected river basins (Marchenko, 

2001; Marchenko et al., 2007) and also at regional 

scale for the entire Tien Shan permafrost domain. 

Regional-scale simulation used a multi-layered 

numerical soil model, including the latent heat of 

fusion, with snow cover and vegetation having 

time-dependent thermal properties. The model uses 

soil extending down to a depth at least of 100 m 

(without horizontal fluxes), and takes into account 

convective cooling within coarse debris and underlying 

soils (Marchenko, 2001). An international team 

of experts is currently working on a unified permafrost 

map of Central Asia (Lai et al., 2006), including the 

Altai Mountain region. Figure 5 shows the first attempt 

to simulate the Altai’s spatially distributed altitudinal 

permafrost, with a 5-km grid size. 

Permafrost in Global/RCMs 

Until recently the transient response of permafrost to 

projected climate change has usually been modelled 

outside of GCMs, using GCM results only to force 

surface conditions, in what Nicolsky et al. (2007) call 

a ‘post-processing approach’, primarily because the 

coarse subsurface resolution within GCMs did not 

adequately represent permafrost processes. The 

representation of the soil column within early GCM 

land surface schemes did not include freezing and 

thawing processes, and most recent implementations 

include few soil layers and a soil column of less than 

10 m (e.g. Li and Koike, 2003; Lawrence and Slater, 

2005; Saha et al., 2006; Saito et al., 2007; Sushama 

et al., 2007). Christensen and Kuhry (2000) and 

Stendel and Christensen (2002) used RCM- and 

GCM-derived surface temperature indices to model 

active-layer thickness using the Stefan equation, and 

permafrost distribution using the Frost Number model, 

Figure 5 (A) Location map for the Altai Mountains; (B) modelled 

permafrost distribution within the northwest part of the region. 

while Stendel et al. (2007) evaluated permafrost 

conditions by driving the Kudryavtsev model with 

GCM output downscaled to RCM resolution. Li and 

Koike (2003) and Yi et al. (2006) have developed 

schemes to improve the accuracy of frost line or 

active-layer thickness estimates with coarsely layered 

soils grids using modified forms of the Stefan 

equation. 

Lawrence and Slater (2005) modelled transient 

permafrost evolution directly within the Community 

Climate System Model GCM (Collins et al., 2006) and 

Community Land Model Version 3 (CLM3) (Oleson 

et al., 2004). Their results generated significant 

discussion within the permafrost community (Burn 

and Nelson, 2006; Lawrence and Slater, 2006; Delisle, 

2007; Yi et al., 2007) for reasons largely related to the 

shallow (3.43 m) soil profile (see following section on 

depth, memory and spin-up), although Yi et al. (2007) 

also demonstrate the importance of the surface organic 

layer. Subsequent work on the geothermal component 

of CLM3 (Mölders and Romanovsky, 2006; Alexeev 

et al., 2007; Nicolsky et al., 2007) has clarified many 

issues involved in accounting for the annual permafrost 

regime. While a modified scheme has been 

evaluated against long-term monitoring data (Mölders 

and Romanovsky, 2006), spatial results using the 

modified scheme at global or regional scale have not 

yet appeared. 


DOI: 10.1002/ppp

At present the most realistic representation of 

transient permafrost dynamics within spatial models is 

within externally forced ‘post-processed’ models 

which incorporate high-resolution finite-element or 

finite-difference, numerical heat flow models (e.g. 

Oelke and Zhang, 2004; Zhang Y. et al., 2005, 2006). 

The application of RCM/GCM output to the 

physics-based modelling of permafrost is especially 

difficult in mountain areas. Because of the poor 

representation of steep topography on coarse grids, 

strong differences between simulated and measured 

climate in those areas exist. Salzmann et al. (2006, 

2007) and Noetzli et al. (2007) have described a 

method that allows downscaling RCM/GCM results in 

mountain areas and their application to energybalance 

and three-dimensional temperature modelling 

in steep bedrock. 

DISCUSSION AND FUTURE DIRECTIONS 

Model Uncertainty and Data Uncertainty 

Permafrost models that operate on broad (circumpolar/continental) 

geographical scale are the most 

appropriate tools for providing descriptions of 

climate-permafrost interactions over the terrestrial 

Arctic. However, their spatial resolution and accuracy 

are limited by availability of data characterising the 

spatial heterogeneity of many important processes 

controlling the ground thermal regime. Shiklomanov 

et al. (2007) found large differences between spatially 

modelled active-layer fields produced by various 

small-scale permafrost models, due primarily to 

differences in the models’ approaches to characterisation 

of largely unknown spatial distributions of 

surface (vegetation, snow) and subsurface (soil 

properties, soil moisture) conditions. Further, Anisimov 

et al. (2007) found the differences in global 

baseline climate datasets (air temperature, precipitations), 

widely used for forcing small-scale permafrost 

models, can translate into uncertainty of up to 20 

per cent in estimates of near-surface permafrost area, 

which is comparable to the extent of changes projected 

for the current century. 

Lower Boundary and Initial Conditions — 

Depth, Memory and Spin-up 

Ground temperature is largely controlled by changes 

at the surface, with change lagged and damped by 

diffusion at depth. The depth to which numerical 

models simulate temperature is limited by the 

relationship between the thermal properties of the 


ground and the size of time step and grid spacing (and 

as determined by the limitations of the model). 

One-dimensional permafrost model studies typically 

specify a modelled soil depth of 20 m or more, in order 

to capture the annual ground temperature cycle, with 

deeper profiles specified when the long-term evolution 

of the ground thermal regime is being evaluated. 

The medium- and long-term fate of permafrost is 

determined by changes at depth, such as the 

development of supra-permafrost taliks. Truncating 

the ground temperature profile at too shallow a depth 

will introduce errors as the deep profile influences the 

thermal regime of the near surface, with errors 

accumulating as the temperature change at the surface 

penetrates to the base of the profile. 

Alexeev et al. (2007) suggest that in general the 

lower boundary should be specified so that the total 

soil depth is much greater than the damping length for 

the timescale of interest. They suggest that the 

timescale of maximum error is about two years for 

a 4 m-deep soil layer, or about 200 years for a 

30 m-deep grid, although their analysis ignores the 

effect of latent heat. In a comparison with an 

equilibrium model (a situation with no memory at 

all) and a numerical model that accounts for latent 

heat, Riseborough (2007) showed that the long-term 

mean annual temperature at the base of the active layer 

could be accurately predicted under transient conditions 

except where a talik was present. Applying this 

result to the analysis of Alexeev et al. (2007), they 

likely underestimate the magnitude of errors due to a 

shallow base, as thaw to the base does not imply the 

disappearance of permafrost, but does imply the 

development of a talik, and that permafrost is no 

longer sustainable under the changing climate. 

The inclusion of a deep temperature profile for the 

modelled space introduces the additional challenge of 

estimating a realistic initial condition. While this can 

be established using field temperature data, this 

approach is impractical in spatial modelling; where 

no data are available, an initial temperature profile is 

usually established by an equilibration or ‘spin-up’ 

procedure, running the model through repeated annual 

cycles with a stationary surface climate, until an 

equilibrium ground temperature profile develops. 

Depending on the profile used to initiate the spin-up 

procedure, equilibration of deep profiles typically 

requires hundreds of cycles, although as Lawrence and 

Slater (2005) and others have noted, the initial profile 

has little effect if the profile is very shallow. In a 

regional permafrost model, with a 120 m-deep 

temperature profile Ednie et al. (2008) found that 

permafrost profiles equilibrated using twentieth 

century climate data did not adequately reproduce 


DOI: 10.1002/ppp


the essential characteristics of the current regime in 

the Mackenzie Valley, Northwest Territories, Canada, 

in particular the presence of deep, nearly isothermal 

(disequilibrium) permafrost. They initiated their 

model with a spin-up to equilibrium for the year 

1721, followed by a surface history combining local 

palaeo-climatic reconstructions and the instrumental 

record. For a national scale model, Zhang Y. et al. 

(2005, 2006) began their long-term simulations 

assuming equilibrium in 1850, with climate data for 

the 1850–1900 period estimated by backward linear 

extrapolation from twentieth century climatic data. 

Initial modelling of permafrost distribution in the 

Mackenzie Valley (Northwest Territories, Canada) 

used the TTOP model (Wright et al., 2003), predicting 

mean annual ground temperature and permafrost 

thickness. The TTOP model was used to improve 

efficiency when modelling the transient evolution of 

permafrost in this environment using a onedimensional 

finite-element model (Duchesne et al., 

2008). First, the accuracy of the TTOP model for 

estimating equilibrium conditions allowed for a 

reasonable first estimate of the initial ground 

temperature profile, minimising the time required 

for model equilibration. Second, the equilibrium 

permafrost thickness estimated using the TTOP model 

was used to establish the depth of the bottom of the 

grid. 

Modelling Permafrost and Environmental 

Change 

Future refinements to modelling of permafrost 

response to climate change projections will require 

consideration of the interaction between permafrost, 

snow cover, vegetation and other environmental 

factors at timescales ranging from decadal to 

millennial. To a significant degree, the position of 

treeline controls the geographic distribution of snow 

density (Riseborough, 2004), so that changes in 

permafrost distribution, migration of treeline and 

changes in snow cover properties in the forest-tundra 

transition zone will be highly inter-dependent. At 

much longer timescales, the distribution of peatlands 

depends on the relative rates of carbon accumulation 

and depletion in the soil, with the zone of peak soil 

carbon and peatland distribution close to the position 

of the 08C mean annual ground temperature isotherm 

(Swanson et al., 2000). Changing ground temperatures 

under climate warming will alter the distribution of 

peatlands, thereby altering local permafrost distribution. 

These changes will influence surface and 

subsurface physical, thermal and hydraulic properties, 

which are often currently assumed to remain 

unchanged, even in a future climate. 

N Factors and Parameterisation 

N factors expressed as a ratio (equation 8) or the layers 

of the Kudryavtsev model (which function by 

diffusive extinction) may not adequately express the 

empirical relationships among the multitude of 

processes they subsume and the underlying system 

behaviour (as opposed to model behaviour). The 

functional form in which parameters encapsulate 

complex processes in simple models will influence 

the behaviour of the model, especially where models 

are used for predictions beyond the scope in which the 

parameters were derived. Unless there is a strong 

theoretical basis for the functional form of the 

parameterisation, it may be better to express the 

relationship in a form dictated by empirical results; 

alternate forms include differences or multi-parameter 

linear or non-linear relationships. 

CONCLUSION 

The fate of permafrost under a changing climate has 

been a concern since the earliest GCM results 

demonstrated the magnitude of the problem. Current 

permafrost transient modelling efforts generally 

follow two approaches: incorporation of permafrost 

dynamics directly into GCM surface schemes and 

increasingly sophisticated regional, national and 

global models forced by GCM output. Although 

computational cost is becoming less critical as 

computer technology advances, these two approaches 

will continue to evolve in tandem, and are unlikely to 

merge. While Stevens et al. (2007) demonstrate the 

importance of the deep geothermal regime on heat 

storage under a changing climate, the influence of 

permafrost conditions on the evolving GCM climate 

and the computational requirements of a multi-layer 

ground thermal scheme will likely be balanced with a 

level of complexity that is less than can be achieved in 

detailed regional models. Once this balance is 

achieved, the relationship between GCM-based and 

post-processed models will be equivalent to the 

relationship between global and regional climate 

models. 

Historically, spatial models of permafrost in arctic 

lowlands and mountain terrain have developed 

following different principles. Permafrost was initially 

studied in Arctic lowland areas, mainly because of 

human development in these regions. Both numerical 

and analytical solutions were developed early, both of 


DOI: 10.1002/ppp

which were easy to apply to spatial models as GIS and 

computer power developed. Permafrost in mountains 

was recognised as an engineering and scientific topic 

much later (1970s), and the spatial heterogeneity of 

factors influencing permafrost led to the development 

of empirical concepts. A trend apparent today is the 

merging of the concepts developed in Arctic lowland 

regions in mountain environments. This is now 

possible with the increasing power of computer 

hardware and software, but the treatment of sub-grid 

heterogeneity (such as the effect of topography) over 

continental or hemispheric areas remains a major 

scientific challenge. 

ACKNOWLEDGEMENTS 

The authors would like to thank Caroline Duchesne 

and two anonymous reviewers whose comments were 

useful in revising the paper. Yu Zhang and Caroline 

Duchesne provided graphical examples of recent work 

for inclusion as figures. The forbearance of the journal’s 

editors is gratefully acknowledged. 

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DOI: 10.1002/ppp

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